How do I find all of the roots, rational and irrational, of the quartic polynomial below?

Question

f(x) = x⁴ - x² + x + 4

Answer 1

Make the substitution y = x² to get

y² - y + 4 = 0

Solve for y by quadratic formula.

The roots of this quadratic are complex.

To get the roots you will have to convert the complex numbers into polar co-ordinates and extract the

square root. This will give you 4 roots as required.

The roots of the quadratic are (1±sqrt(-15))/2.

The polar form is 2e^±iθθ where θ = arctan(sqrt15))

The square root then is ±√2(e^±iθ/2)

That should do it.